Interpret the wave equation

The given longitudinal wave is $\xi = 10\sin 2\pi\left(nt-\frac{x}{\lambda}\right)\text{ cm}$ where $\xi$ is the particle displacement.

Comparing with the standard form $\xi = a\sin 2\pi\left(ft-\frac{x}{\lambda}\right),$ we get:

• Amplitude: $a=10\text{ cm}$
• Frequency: $f=n$
• Wavelength: $\lambda=?$

Maximum particle velocity

Particle velocity is $v_p=\frac{\partial \xi}{\partial t}$

So, $v_p=10\cdot 2\pi n\cos 2\pi\left(nt-\frac{x}{\lambda}\right)$

Hence maximum particle velocity is $v_{p,\max}=10\cdot 2\pi n=20\pi n\text{ cm/s}$

Wave velocity

Wave speed is $v=f\lambda=n\lambda$

Use the given condition

According to the question, $v_{p,\max}=4v$

Therefore, $20\pi n=4(n\lambda)$

Cancel $n$: $20\pi=4\lambda$

So, $\lambda=5\pi$

Match with options

$\lambda=5\pi$

Therefore the correct option is B.